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This paper formalizes the stepping stone problem introduced Ladoucer and Rebenstock [G] to the setting of simple graphs. This paper considers the set of functions from the vertices of our graph to N, which assign a fixed number of 1's to some vertices and assign higher numbers to other vertices by adding up their neighbors' assignments. The stepping stone solution is defined as an element obtained from the argmax of the maxima of these functions, and the maxima as its growth. This work is organized into work on the bounded and unbounded degree graph cases. In the bounded case, sufficient conditions are obtained for superlinear and sublinear stepping stone solution growth. Furthermore this paper demonstrates the existence of a basis of graphs which characterizes superlinear growth. In the unbounded case, properly sublinear and superlinear stepping stone solution growth are obtained. [ABSTRACT FROM AUTHOR]

A reaction system is a computational model motivated by the functioning of living cells. In previous studies, Ehrenfeucht et al. have classified the functions specified by minimal reaction systems. Several studies have focused on the reaction system ranks of functions specified by minimal reaction systems, where the rank pertains to the minimum size of a specifying reaction system. This paper focuses on reaction system rank for a class of union-additive functions specified by minimal reaction systems introduced by Salomaa, which are closed under composition. More precisely, we establish a sufficient condition for the equality of reaction system ranks for functions belonging to the class. Then we study the reaction system ranks of such functions with signature sizes one and two, as well as those with one-to-one signatures, specifically over ternary and quaternary alphabets. Lastly, we establish a lower upper bound of reaction system ranks of such functions with one-to-one signatures for | S | ≥ 5. [ABSTRACT FROM AUTHOR]

Linear codes play a crucial role in various fields of engineering and mathematics, including data storage, communication, cryptography, and combinatorics. Minimal linear codes, a subset of linear codes, are particularly essential for designing effective secret sharing schemes. In this paper, we introduce several classes of minimal binary linear codes by carefully selecting appropriate Boolean functions. These functions belong to a renowned class of Boolean functions, namely, the general Maiorana–McFarland class. We employ a method first proposed by Ding et al. (IEEE Trans Inf Theory 64(10):6536–6545, 2018) to construct minimal codes violating the Ashikhmin–Barg bound (wide minimal codes) by using Krawtchouk polynomials. The lengths, dimensions, and weight distributions of the obtained codes are determined using the Walsh spectrum distribution of the chosen Boolean functions. Our findings demonstrate that a vast majority of the newly constructed codes are wide minimal. Furthermore, our proposed codes exhibit a significantly larger minimum distance, in some cases, compared to some existing similar constructions. Finally, we address this method, based on Krawtchouk polynomials, more generally, and highlight certain generic properties related to it. These general results offer insights into the scope of this approach. [ABSTRACT FROM AUTHOR]

The method of hypergraph containers, which was introduced several years ago by Balogh, Morris, and Samotij, and independently by Saxton and Thomason, has proved to be an extremely useful tool in the study of various monotone graph properties. In particular, a fairly straightforward application of this technique allows one to locate, for each non-bipartite graph H, the threshold at which the distribution of edges in a typical H-free graph with a given number of edges undergoes a transition from 'random-like' to 'structured'. On the other hand, for nonmonotone hereditary graph properties, the standard version of this method does not allow one to establish even the existence of such a threshold. In this paper we introduce a refinement of the container method that takes into account the asymmetry between edges and non-edges in a sparse member of a hereditary graph property. As an application, we determine the approximate structure of a typical graph with n vertices, m edges, and no induced copy of the 4-cycle, for each function m D m.n/ satisfying n4=3.log n/4 6 m n2. We show that almost all such graphs G have the following property: the vertex set of G can be partitioned into an 'almost independent' set (a set with o.m/ edges) and an 'almost-clique' (a set inducing a subgraph with density 1-a The lower bound on m is optimal up to a polylogarithmic factor, as standard arguments show that if n then almost all such graphs are 'random-like'. As a further consequence, we deduce that the random graph G.n; p/ conditioned to contain no induced 4-cycles undergoes phase transitions at p=n -2/3+o(1) p=n -2/3+o(1). [ABSTRACT FROM AUTHOR]

This paper proposes explicit solutions for the algebraic Riccati matrix equation. For single‐input systems in controllable canonical form, the explicit Hermitian solutions of the non‐homogeneous Riccati equation are obtained using the entries of the system matrix, the closed‐loop system matrix, and the weighting matrix. The unknown entries of the closed‐loop system matrix are solved by scalar quadratic equations. For a homogeneous Riccati equation with a zero weighting matrix, the explicit solutions are proposed analytically in terms of the system eigenvalues. The advantages of the explicit solutions are threefold: first, if the system is controllable, the solution is directly given and the invariant subspaces of the Hamiltonian matrix are not required; second, if the system is near singularity, the explicit solution has higher numerical precision compared with the solution computed by numerical algorithms; third, for a real system in the controllable canonical form, the non‐negativity can be analysed for the explicit almost stabilizing solution. [ABSTRACT FROM AUTHOR]

In problems such as variable selection and graph estimation, models are characterized by Boolean logical structure such as the presence or absence of a variable or an edge. Consequently, false-positive error or false-negative error can be specified as the number of variables/edges that are incorrectly included or excluded in an estimated model. However, there are several other problems such as ranking, clustering, and causal inference in which the associated model classes do not admit transparent notions of false-positive and false-negative errors due to the lack of an underlying Boolean logical structure. In this paper, we present a generic approach to endow a collection of models with partial order structure, which leads to a hierarchical organization of model classes as well as natural analogs of false-positive and false-negative errors. We describe model selection procedures that provide false-positive error control in our general setting, and we illustrate their utility with numerical experiments. [ABSTRACT FROM AUTHOR]

The q-color Ramsey number of a k-uniform hypergraph G, denoted r(G; q), is the minimum integer N such that any coloring of the edges of the complete k-uniform hypergraph on N vertices contains a monochromatic copy of G. The study of these numbers is one of the most central topics in combinatorics. One natural question, which for triangles goes back to the work of Schur in 1916, is to determine the behavior of r(G; q) for fixed G and q tending to infinity. In this paper, we study this problem for 3-uniform hypergraphs and determine the tower height of r(G; q) as a function of q. More precisely, given a hypergraph G, we determine when r(G; q) behaves polynomially, exponentially or double exponentially in q. This answers a question of Axenovich, Gyárfás, Liu and Mubayi. [ABSTRACT FROM AUTHOR]

A matching of a graph is a set of its edges that pairwise do not have common vertices. An important parameter of graphs, which is used in mathematical chemistry, is the Hosoya index, defined as the number of their matchings. Previously, the problems of maximizing this index were considered and completely solved for -vertex trees with two, three and four leaves for any sufficiently large . In the present paper, a similar problem is completely solved for 5-leaved trees with and for 6-leaved trees with . [ABSTRACT FROM AUTHOR]

A set C of vertices in a graph G = (V, E) is an identifying code if it is dominating and any two vertices of V are dominated by distinct sets of codewords. This paper presents a survey of Iiro Honkala's contributions to the study of identifying codes with respect to several aspects: complexity of computing an identifying code, combinatorics in binary Hamming spaces, infinite grids, relationships between identifying codes and usual parameters in graphs, structural properties of graphs admitting identifying codes, and number of optimal identifying codes. [ABSTRACT FROM AUTHOR]

Operations in the theory of graphs has a substantial inuence in the analytical and factual dimensions of the domain. In the realm of chemical graph theory, topological descriptor serves as a comprehensive graph invariant linked with a specific molecular structure. The study on the Sombor index is initiated recently by Ivan Gutman. The triangle parallel graph comprises of the edges of subdivision graph along with the edges of the original graph. In this paper, we make use of combinatorial inequalities related with the vertices, edges and the neighborhood concepts as well as the other topological descriptors in the computations for the determination of bounds of Sombor index for certain corona products involving the triangle parallel graph. [ABSTRACT FROM AUTHOR]

Catalan sequence initially introduced by Euler but named after Eugene Catalan has found abundant applications in the fields of probability theory, number theory, analysis, combinatorics etc. Recently ˙Ilkhan formulated the matrix associated with Catalan numbers and studied its domain in c0(C~) and c(C~). In this paper, by employing Catalan matrix defined by Catalan sequence and convergence via ideal, we establish Catalan sequence spaces c I 0 (C~), c I (C~) and ℓ I ∞(C~), study the properties of these corresponding spaces and establish inclusion relations. [ABSTRACT FROM AUTHOR]

The text is a foreword for a special issue of the Journal of Information Science & Engineering on Combinatorial Mathematics and Computational Theory. It explains that these fields have a significant impact on computer science, mathematics, and science in general. Combinatorial Mathematics involves the study of discrete objects and has applications in various areas, while Computational Theory focuses on studying algorithms and their efficiency. The text also mentions a symposium that was held to promote academic exchanges and advance research in these fields. The special issue includes four papers on topics such as opinion optimization, local diagnosis, enumerating furthest pairs, and determining the 2-tuple total domination number of a graph. The authors express gratitude to the reviewers and authors for their contributions to the special issue. [Extracted from the article]

The power graph P (Z n) of Z n for a finite cyclic group Z n is a simple undirected connected graph such that two distinct nodes x and y in Z n are adjacent in P (Z n) if and only if x ≠ y and x i = y or y i = x for some non-negative integer i . In this article, we find the Laplacian eigenvalues of P (Z n) and show that P (Z n) is Laplacian integral (integer algebraic connectivity) if and only if n is either the product of two distinct primes or a prime power. That answers a conjecture by Panda, Graphs and Combinatorics, (2019). [ABSTRACT FROM AUTHOR]

A μ-way (v,k,t) trade T of volume m consists of μ pairwise disjoint collections Ti,...,Tμ, each of m blocks of size k such that for every t-subset of a v-set V, the number of blocks containing this t-subset is the same in each Ti for 1 ≤ i ≤ μ. If any t-subset of the v-set V occurs at most once in each Ti for 1 ≤ i ≤ μ, then T is called a μ-way (v, k, t) Steiner trade. In 2016, it was proved that there exists a 3-way (v, k, 2) Steiner trade of volume m when 12(k - 1) ≤ m for each k. Here we improve the lower bound to 8(k - 1) for even k, by using a recursive construction. [ABSTRACT FROM AUTHOR]

This article studies the bidimensional versions of four number sequences: balancing, Lucasbalancing, Lucas-cobalancing and cobalancing. Some of the identities and generating functions were presented. [ABSTRACT FROM AUTHOR]

A family F of sets is said to be t -intersecting if for any A; B 2 F . The seminal Complete Intersection Theorem of Ahlswede and Khachatrian (1997) gives the maximal size f .n; k; t of a t -intersecting family of k-element subsets of OEn D together with a characterisation of the extremal families, solving a longstanding problem of Frankl. The forbidden intersection problem, posed by Erdos and Sós in 1971, asks for a determination of the maximal size g.n; k; t / of a family F of k-element subsets of OEn such that jA \ Bj ¤ for any A, B F. [ABSTRACT FROM AUTHOR]

Feynman diagrams represent one of the most powerful and fascinating tools developed in theoretical physics in the last century. Introduced within the framework of quantum electrodynamics as a suitable method for computing the amplitude of a physical process, they rapidly became a fundamental mathematical object in quantum field theory. However, their abstract nature seems to suggest a wider usage, which actually exceeds the physical context. Indeed, as mathematical objects, they could simply be considered graphs that depict not only physical quantities but also biological or economic entities. We survey the analytical and algebraic properties of such diagrams to understand their utility in several areas of science, eventually providing some examples of recent applications. [ABSTRACT FROM AUTHOR]

The issue of counting independent sets of a graph, G, represented as i (G) , is a significant challenge within combinatorial mathematics. This problem finds practical applications across various fields, including mathematics, computer science, physics, and chemistry. In chemistry, i (G) is recognized as the Merrifield–Simmons (M-S) index for molecular graphs, which is one of the most relevant topological indices related to the boiling point in chemical compounds. This article introduces an innovative algorithm designed for tallying independent sets within grid-like structures. The proposed algorithm is based on the 'branch-and-bound' technique and is applied to compute i (G m , n) for a square grid formed by m rows and n columns. The proposed approach incorporates the widely recognized vertex reduction rule as the basis for splitting the current subgraph. The methodology involves breaking down the initial grid iteratively until outerplanar graphs are achieved, serving as the 'basic cases' linked to the leaf nodes of the computation tree or when no neighborhood is incident to a minimum of five rectangular internal faces. The time complexity of the branch-and-bound algorithm speeds up the computation of i (G m , n) compared to traditional methods, like the transfer matrix method. Furthermore, the scope of the proposed algorithm is more general than the algorithms focused on grids since it could be applied to process general mesh graphs. [ABSTRACT FROM AUTHOR]

Many important problems are closely related to the zeros of certain polynomials derived from combinatorial objects. The aim of this paper is to observe some results and applications for the Hurwitz stability of polynomials in combinatorics and study other related problems. We first present a criterion for the Hurwitz stability of the Turán expressions of recursive polynomials. In particular, it implies the q -log-convexity or q -log-concavity of the original polynomials. We also give a criterion for the Hurwitz stability of recursive polynomials and prove that the Hurwitz stability of any palindromic polynomial implies its semi- γ -positivity, which illustrates that the original polynomial with odd degree is unimodal. In particular, we get that the semi- γ -positivity of polynomials implies their parity-unimodality and the Hurwitz stability of polynomials implies their parity-log-concavity. Those results generalize the connections between real-rootedness, γ -positivity, log-concavity and unimodality to Hurwitz stability, semi- γ -positivity, parity-log-concavity and parity-unimodality (unimodality). As applications of these criteria, we derive some Hurwitz stability results occurred in the literature in a unified manner. In addition, we obtain the Hurwitz stability of Turán expressions for alternating run polynomials of types A and B and the Hurwitz stability for alternating run polynomials defined on a dual set of Stirling permutations. Finally, we study a class of recursive palindromic polynomials and derive many nice properties including Hurwitz stability, semi- γ -positivity, non- γ -positivity, unimodality, strong q -log-convexity, the Jacobi continued fraction expansion and the relation with derivative polynomials. In particular, these properties of the alternating descents polynomials of types A and B can be implied in a unified approach. [ABSTRACT FROM AUTHOR]

We reconsider the classical equality 0.999 . . . = 1 with the tool of circular words, that is, finite words whose last letter is assumed to be followed by the first one. Such circular words are naturally embedded with algebraic structures that enlight this problematic equality, allowing it to be considered in Q rather than in R. We comment early history of such structures, that involves English teachers and accountants of the first part of the 18th century, who appear to be the firsts to assert the equality 0.999... = 1. Their level of understanding show links with Dubinsky et al.'s apos theory in mathematics education. Eventually, we rebuilt the field Q from circular words, and provide an original proof of the fact that an algebraic integer is either an integer or an irrational number. [ABSTRACT FROM AUTHOR]

We analyzed the symptoms composition of Interstitial Cystitis (IC), the regularity of the evolution of symptoms before and after treatment, and the visualization of the community network, to provide a reference for clinical diagnosis and treatment of Interstitial Cystitis. Based on the outpatient electronic case data of 552 patients with Interstitial Cystitis, we used a complex network community discovery algorithm, directed weighted complex network, and Sankey map to mine the data of the symptoms composition of Interstitial Cystitis, the evolution of symptoms before and after treatment and the visualization of the community network, to analyze the epidemiological characteristics of interstitial cystitis symptoms in the real world. By the community division of the complex network of interstitial cystitis symptoms, We finally obtained three core symptom communities. Among them, symptom community A (bladder‐related symptoms) is the symptom community with the highest proportion of nodes (60.00%) in the complex network of Interstitial Cystitis, symptom community B (non‐bladder‐related symptoms 1) ranks second (32.00%) in a complex network of Interstitial Cystitis, and symptom community C (non‐bladder‐related symptoms 2) has the lowest proportion (8.00%). There is a complex evolutionary relationship between the symptoms of Interstitial Cystitis before and after treatment. Among the single symptoms before and after treatment, the decreased rate of Day frequency is 93.22%, and the reduced urgency rate is 93.07%. The decline rate of Nocturia was 82.33%. From the perspective of different communities, the overall symptoms of symptom community A decreased by 34.39% after treatment, the general symptoms of symptom community B decreased by 35.37%, and the prevalent symptoms of symptom community C decreased by 71.43%. In the case of using diet regulation treatment to treat bladder pain, the cure rate of bladder pain can reach 22.67%. The cure rate of burning in bladders can get 15.38% with Percutaneous Sacral neuromodulation, 96.95% with diet regulation treatment, and 100% with Percutaneous Sacral neuromodulation. When using behavioral physiotherapy to treat bladder pain, 3.57% of the patient's symptoms change to bladder discomfort; 4% of the patient's symptoms change to bladder discomfort when using oral medicine to treat bladder pain.Symptom research methods based on community findings can effectively explore the rule of symptom outcome of Interstitial Cystitis before and after treatment, and the results are highly interpretable by professionals. The cover image is based on the Original Article Research on symptoms composition, time series evolution, and network visualisation of interstitial cystitis based on complex network community discovery algorithm by Lei Pang et al., https://doi.org/10.1049/syb2.12083 [ABSTRACT FROM AUTHOR]

We study the combinatorics of modular flats of oriented matroids and the topological consequences for their Salvetti complexes. We show that the natural map to the localized Salvetti complex at a modular flat of corank one is what we call a poset quasi-fibration – a notion derived from Quillen's fundamental Theorem B from algebraic K-theory. As a direct consequence, the Salvetti complex of an oriented matroid whose geometric lattice is supersolvable is a K(\pi,1)-space – a generalization of the classical result for supersolvable hyperplane arrangements due to Falk, Randell and Terao. Furthermore, the fundamental group of the Salvetti complex of a supersolvable oriented matroid is an iterated semidirect product of finitely generated free groups – analogous to the realizable case. Our main tools are discrete Morse theory, the shellability of certain subcomplexes of the covector complex of an oriented matroid, a nice combinatorial decomposition of poset fibers of the localization map, and an isomorphism of covector posets associated to modular elements. We provide a simple construction of supersolvable oriented matroids. This gives many non-realizable supersolvable oriented matroids and by our main result aspherical CW-complexes. [ABSTRACT FROM AUTHOR]

The notion of an extriangulated category gives a unification of existing theories in exact or abelian categories and in triangulated categories. In this article, we develop Auslander–Reiten theory for extriangulated categories. This unifies Auslander–Reiten theories developed in exact categories and triangulated categories independently. We give two different sets of sufficient conditions on the extriangulated category so that existence of almost split extensions becomes equivalent to that of an Auslander–Reiten–Serre duality. We also show that existence of almost split extensions is preserved under taking relative extriangulated categories, ideal quotients, and extension-closed subcategories. Moreover, we prove that the stable category \underline {\mathscr {C}} of an extriangulated category \mathscr {C} is a \tau-category (see O. Iyama [Algebr. Represent. Theory 8 (2005), pp. 297–321]) if \mathscr {C} has enough projectives, almost split extensions and source morphisms. This gives various consequences on \underline {\mathscr {C}}, including Igusa–Todorov's Radical Layers Theorem (see K. Igusa and G. Todorov [J. Algebra 89 (1984), pp. 105–147]), Auslander–Reiten Combinatorics on dimensions of Hom-spaces, and Reconstruction Theorem of the associated completely graded category of \underline {\mathscr {C}} via the complete mesh category of the Auslander–Reiten species of \underline {\mathscr {C}}. Finally we prove that any locally finite symmetrizable \tau-quiver (=valued translation quiver) is an Auslander–Reiten quiver of some extriangulated category with sink morphisms and source morphisms. [ABSTRACT FROM AUTHOR]

Mathematics is often presented as a neatly polished finished product, yet its development is messy and often full of mis-steps that could have been avoided with hindsight. An experience with a puzzle illustrates this conflict. The puzzle asks for the probability that a group of four and a group of two are seated adjacently within a hundred seats, and is solved using combinatorics techniques. [ABSTRACT FROM AUTHOR]

Symmetry is often regarded as an integral aspect about aesthetics. This motivates the pursuit of interdisciplinary studies based on the use of subjects in mathematics concerned with symmetry in conjunction with aesthetics. What is referred to as a symmetric function in the field of algebraic combinatorics is an abstraction based on polynomials that exhibit a symmetric property, and this leads us to pursue an algebraic combinatorics-inspired exploration based on aesthetics. In particular, we use different bases and transitions between them to create aesthetically pleasing visualizations of symmetric functions. We see that these visualizations in turn raise new and interesting questions. [ABSTRACT FROM AUTHOR]